What does pi look like? Pi number - interesting information
1. Pi was known already 4 thousand years ago
In the Babylonian kingdom the number was 25/8 (or in decimal form 3.125), in Ancient Egypt - 256/81 (about 3.1605), in Ancient India - 339/108 (about 3.1389).
The ancient Greek scientist Archimedes was the first to propose using a method for calculating the number π, which determines the dependence of the circumference of a circle on its diameter. This is how the world received the first approximate value of the number π, equal to ~22/7 (about 3.14286).
2. The number Pi is carved on the tombstone of a Dutch mathematician
At the beginning of the 17th century, the Dutch scientist Ludolf van Zeijlen spent ten years calculating π with an accuracy of 35 digits using series. It is not surprising that the scientist bequeathed that the number π be carved with this accuracy on his tombstone. In honor of van Zeijlen, pi was also sometimes called the "Ludolff number".
3. There are 22,400,000,000,000 known first digits of Pi
With the development of civilization, the need for accuracy of calculations grew, which also required increasing the accuracy of the number π. True, forty decimal places are enough, for example, to calculate a circle the size of a Galaxy, and with an accuracy of one ten millionth of a millimeter!
Moreover, the accuracy of calculating π, which is realized by modern computers, exceeds all conceivable needs. But the race for accuracy apparently cannot be stopped. In 1973, π was calculated to an accuracy of one million digits, and in 2011, Japanese engineer Shigeru Kondo calculated π to an accuracy of 10,000,000,000 (trillion) digits. His record was broken in 2016 by Czech physicist Peter Trueb - 22,400,000,000,000 characters. Both used a program to calculate
If you compare circles of different sizes, you will notice the following: the sizes of different circles are proportional. This means that when the diameter of a circle increases by a certain number of times, the length of this circle also increases by the same number of times. Mathematically this can be written like this:
| C 1 | C 2 | ||
| = | |||
| d 1 | d 2 | (1) |
where C1 and C2 are the lengths of two different circles, and d1 and d2 are their diameters.
This relationship works in the presence of a coefficient of proportionality - the constant π already familiar to us. From relation (1) we can conclude: the length of a circle C is equal to the product of the diameter of this circle and a proportionality coefficient π independent of the circle:
C = π d.
This formula can also be written in another form, expressing the diameter d through the radius R of a given circle:
С = 2π R.
This formula is precisely the guide to the world of circles for seventh graders.
Since ancient times, people have tried to establish the value of this constant. For example, the inhabitants of Mesopotamia calculated the area of a circle using the formula:
Where does π = 3 come from?
In ancient Egypt, the value for π was more precise. In 2000-1700 BC, a scribe called Ahmes compiled a papyrus in which we find recipes for solving various practical problems. So, for example, to find the area of a circle, he uses the formula:
| 8 | 2 | |||||
| S | = | ( | d | ) | ||
| 9 |
From what reasons did he arrive at this formula? – Unknown. Probably based on his observations, however, as other ancient philosophers did.
In the footsteps of Archimedes
Which of the two numbers is greater than 22/7 or 3.14?
- They are equal.
- Why?
- Each of them is equal to π.
A. A. Vlasov. From the Examination Card.
Some people believe that the fraction 22/7 and the number π are identically equal. But this is a misconception. In addition to the above incorrect answer in the exam (see epigraph), you can also add one very entertaining puzzle to this group. The task reads: “arrange one match so that the equality becomes true.”
The solution would be this: you need to form a “roof” for the two vertical matches on the left, using one of the vertical matches in the denominator on the right. You will get a visual image of the letter π.
Many people know that the approximation π = 22/7 was determined by the ancient Greek mathematician Archimedes. In honor of this, this approximation is often called the “Archimedean” number. Archimedes managed not only to establish an approximate value for π, but also to find the accuracy of this approximation, namely, to find a narrow numerical interval to which the value π belongs. In one of his works, Archimedes proves a chain of inequalities, which in a modern way would look like this:
| 10 | 6336 | 14688 | 1 | |||||||||
| 3 | < | < | π | < | < | 3 | ||||||
| 71 | 1 | 1 | 7 | |||||||||
| 2017 | 4673 | |||||||||||
| 4 | 2 | |||||||||||
can be written more simply: 3,140 909< π < 3,1 428 265...
As we can see from the inequalities, Archimedes found a fairly accurate value with an accuracy of up to 0.002. The most surprising thing is that he found the first two decimal places: 3.14... This is the value we most often use in simple calculations.
Practical use
Two people are traveling on a train:
- Look, the rails are straight, the wheels are round.
Where is the knock coming from?
- Where from? The wheels are round, but the area
circle pi er square, that’s the square that knocks!
As a rule, they become acquainted with this amazing number in the 6th-7th grade, but study it more thoroughly by the end of the 8th grade. In this part of the article we will present the basic and most important formulas that will be useful to you in solving geometric problems, but to begin with we will agree to take π as 3.14 for ease of calculation.
Perhaps the most famous formula among schoolchildren that uses π is the formula for the length and area of a circle. The first, the formula for the area of a circle, is written as follows:
| π D 2 | |
| S=π R 2 = | |
| 4 |
where S is the area of the circle, R is its radius, D is the diameter of the circle.
The circumference of a circle, or, as it is sometimes called, the perimeter of a circle, is calculated by the formula:
C = 2 π R = π d,
where C is the circumference, R is the radius, d is the diameter of the circle.
It is clear that the diameter d is equal to two radii R.
From the formula for circumference, you can easily find the radius of the circle:
where D is the diameter, C is the circumference, R is the radius of the circle.
These are basic formulas that every student should know. Also, sometimes it is necessary to calculate the area not of the entire circle, but only of its part - the sector. Therefore, we present it to you - a formula for calculating the area of a sector of a circle. She looks like this:
| α | |||
| S | = | π R 2 | |
| 360 ˚ |
where S is the area of the sector, R is the radius of the circle, α is the central angle in degrees.
So mysterious 3.14
Indeed, it is mysterious. Because in honor of these magical numbers they organize holidays, make films, hold public events, write poems and much more.
For example, in 1998, a film by American director Darren Aronofsky called “Pi” was released. The film received many awards.
Every year on March 14 at 1:59:26 a.m., people interested in mathematics celebrate "Pi Day." For the holiday, people prepare a round cake, sit at a round table and discuss the number Pi, solve problems and puzzles related to Pi.
Poets also paid attention to this amazing number; an unknown person wrote:
You just have to try and remember everything as it is - three, fourteen, fifteen, ninety-two and six.
Let's have some fun!
We offer you interesting puzzles with the number Pi. Unravel the words that are encrypted below.
1. π R
2. π L
3. π k
Answers: 1. Feast; 2. File; 3. Squeak.
Number meaning(pronounced "pi") is a mathematical constant equal to the ratio
Denoted by the letter "pi" of the Greek alphabet. Old name - Ludolph number.
What is pi equal to? In simple cases, it is enough to know the first 3 signs (3.14). But for more
complex cases and where greater accuracy is needed, you need to know more than 3 digits.
What is pi? First 1000 decimal places of pi:
3,1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989...
Under normal conditions, the approximate value of pi can be calculated following the steps,
given below:
- Take a circle and wrap the thread around its edge once.
- We measure the length of the thread.
- We measure the diameter of the circle.
- Divide the length of the thread by the length of the diameter. We got the number pi.
Properties of Pi.
- pi- irrational number, i.e. the value of pi cannot be accurately expressed as
fractions m/n, Where m And n are integers. From this it is clear that the decimal representation
pi never ends and it is not periodic.
- pi- transcendental number, i.e. it cannot be the root of any polynomial with integers
coefficients. In 1882, Professor Koenigsbergsky proved the transcendence pi numbers, A
later, professor at the University of Munich Lindemann. The proof has been simplified
Felix Klein in 1894.
- since in Euclidean geometry the area of a circle and the circumference are functions of pi,
that proof of the transcendence of pi put an end to the dispute about the squaring of the circle, which lasted more than
2.5 thousand years.
- pi is an element of the period ring (that is, a computable and arithmetic number).
But no one knows whether it belongs to the ring of periods.
Pi number formula.
- Francois Viet:
- Wallis formula:
- Leibniz series:
- Other rows:
Today is the birthday of Pi, which, on the initiative of American mathematicians, is celebrated on March 14 at 1 hour and 59 minutes in the afternoon. This is connected with a more precise value of Pi: we are all accustomed to considering this constant as 3.14, but the number can be continued as follows: 3, 14159... Translating this into a calendar date, we get 03.14, 1:59.
Photo: AiF/ Nadezhda Uvarova
Professor of the Department of Mathematical and Functional Analysis of South Ural State University Vladimir Zalyapin says that July 22 should still be considered “Pi day”, because in the European date format this day is written as 22/7, and the value of this fraction is approximately equal to the value of Pi .
“The history of the number that gives the ratio of the circumference to the diameter of the circle goes back to ancient times,” says Zalyapin. - Already the Sumerians and Babylonians knew that this ratio does not depend on the diameter of the circle and is constant. One of the first mentions of the number Pi can be found in the texts Egyptian scribe Ahmes(around 1650 BC). The ancient Greeks, who borrowed a lot from the Egyptians, contributed to the development of this mysterious quantity. According to the legend, Archimedes was so carried away by calculations that he did not notice how Roman soldiers took his hometown of Syracuse. When the Roman soldier approached him, Archimedes shouted in Greek: “Don’t touch my circles!” In response, the soldier stabbed him with a sword.
Plato received a fairly accurate value of Pi for his time - 3.146. Ludolf van Zeilen spent most of his life calculating the first 36 decimal places of Pi, and they were engraved on his tombstone after his death."
Irrational and abnormal
According to the professor, at all times the pursuit of calculating new decimal places was determined by the desire to obtain the exact value of this number. It was assumed that Pi was rational and could therefore be expressed as a simple fraction. And this is fundamentally wrong!
The number Pi is also popular because it is mystical. Since ancient times, there has been a religion of worshipers of the constant. In addition to the traditional value of Pi - a mathematical constant (3.1415...), expressing the ratio of the circumference of a circle to its diameter, there are many other meanings of the number. Such facts are interesting. In the process of measuring the dimensions of the Great Pyramid of Giza, it turned out that it has the same ratio of height to the perimeter of its base as the radius of a circle to its length, that is, ½ Pi.
If you calculate the length of the Earth's equator using Pi to the ninth decimal place, the error in the calculations will be only about 6 mm. Thirty-nine decimal places in Pi are enough to calculate the circumference of the circle surrounding known cosmic objects in the Universe, with an error no greater than the radius of a hydrogen atom!
The study of Pi also includes mathematical analysis. Photo: AiF/ Nadezhda Uvarova
Chaos in numbers
According to a mathematics professor, in 1767 Lambert established the irrationality of the number Pi, that is, the impossibility of representing it as a ratio of two integers. This means that the sequence of decimal places of Pi is chaos embodied in numbers. In other words, the “tail” of decimal places contains any number, any sequence of numbers, any texts that were, are and will be, but it’s just not possible to extract this information!
“It is impossible to know the exact value of Pi,” continues Vladimir Ilyich. - But these attempts are not abandoned. In 1991 Chudnovsky achieved a new 2260000000 decimal places of the constant, and in 1994 - 4044000000. After that, the number of correct digits of Pi increased like an avalanche.”
Chinese holds world record for memorizing Pi Liu Chao, who was able to remember 67,890 decimal places without error and reproduce them within 24 hours and 4 minutes.
About the “golden ratio”
By the way, the connection between “pi” and another amazing quantity - the golden ratio - has never actually been proven. People have long noticed that the “golden” proportion - also known as the number Phi - and the number Pi divided by two differ from each other by less than 3% (1.61803398... and 1.57079632...). However, for mathematics, these three percent are too significant a difference to consider these values identical. In the same way, we can say that the Pi number and the Phi number are relatives of another well-known constant - the Euler number, since the root of it is close to half the Pi number. One half of Pi is 1.5708, Phi is 1.6180, the root of E is 1.6487.
This is only part of the value of Pi. Photo: Screenshot
Pi's birthday
At South Ural State University, the birthday of the constant is celebrated by all teachers and mathematics students. This has always been the case - it cannot be said that interest has appeared only in recent years. The number 3.14 is even welcomed with a special holiday concert!
There are a lot of mysteries among the PIs. Or rather, these are not even riddles, but a kind of Truth that no one has yet solved in the entire history of mankind...
What is Pi? The PI number is a mathematical “constant” that expresses the ratio of the circumference of a circle to its diameter. At first, out of ignorance, it (this ratio) was considered equal to three, which was a rough approximation, but it was enough for them. But when prehistoric times gave way to ancient times (i.e., already historical), the surprise of inquisitive minds knew no bounds: it turned out that the number three very inaccurately expresses this ratio. With the passage of time and the development of science, this number began to be considered equal to twenty-two sevenths.
The English mathematician Augustus de Morgan once called the number PI “... the mysterious number 3.14159... that crawls through the door, through the window and through the roof.” Tireless scientists continued and continued to calculate the decimal places of the number Pi, which is actually a wildly non-trivial task, because you can’t just calculate it in a column: the number is not only irrational, but also transcendental (these are just such numbers that cannot be calculated by simple equations).
In the process of calculating these same signs, many different scientific methods and entire sciences were discovered. But the most important thing is that there are no repetitions in the decimal part of pi, as in an ordinary periodic fraction, and the number of decimal places is infinite. Today it has been verified that there are indeed no repetitions in 500 billion digits of pi. There is reason to believe that there are none at all.
Since there are no repetitions in the sequence of pi signs, this means that the sequence of pi signs obeys the theory of chaos, or more precisely, the number pi is chaos written in numbers. Moreover, if desired, this chaos can be represented graphically, and there is an assumption that this Chaos is intelligent.
In 1965, the American mathematician M. Ulam, sitting at one boring meeting, with nothing to do, began to write the numbers included in pi on checkered paper. Putting 3 in the center and moving counterclockwise in a spiral, he wrote out 1, 4, 1, 5, 9, 2, 6, 5 and other numbers after the decimal point. Along the way, he circled all the prime numbers. Imagine his surprise and horror when the circles began to line up along straight lines!
In the decimal tail of pi you can find any desired sequence of digits. Any sequence of digits in the decimal places of pi will be found sooner or later. Any!
So what? - you ask. Otherwise... Think about it: if your phone is there (and it is), then there is also the phone number of the girl who didn’t want to give you her number. Moreover, there are credit card numbers, and even all the values of the winning numbers for tomorrow's lottery draw. What is there, in general, all lotteries for many millennia to come. The question is how to find them there...
If you encrypt all the letters with numbers, then in the decimal expansion of the number pi you can find all the world literature and science, and the recipe for making bechamel sauce, and all the holy books of all religions. This is a strict scientific fact. After all, the sequence is INFINITE and the combinations in the number PI are not repeated, therefore it contains ALL combinations of numbers, and this has already been proven. And if everything, then ALL. Including those that correspond to the book you have chosen.
And this again means that it contains not only all the world literature that has already been written (in particular, those books that burned, etc.), but also all the books that WILL yet be written. Including your articles on websites. It turns out that this number (the only reasonable number in the Universe!) governs our world. You just need to look at more signs, find the right area and decipher it. This is somewhat akin to the paradox of a herd of chimpanzees hammering away at a keyboard. Given a long enough experiment (you can even estimate the time) they will print all of Shakespeare's plays.
This immediately suggests an analogy with periodically appearing messages that the Old Testament supposedly contains encoded messages to descendants that can be read using clever programs. It is not entirely wise to immediately dismiss such an exotic feature of the Bible; cabalists have been searching for such prophecies for centuries, but I would like to cite the message of one researcher who, using a computer, found words in the Old Testament that there are no prophecies in the Old Testament. Most likely, in a very large text, as well as in the infinite digits of the PI number, it is possible not only to encode any information, but also to “find” phrases that were not originally included there.
For practice, 11 characters after the dot are enough within the Earth. Then, knowing that the radius of the Earth is 6400 km or 6.4 * 1012 millimeters, it turns out that if we discard the twelfth digit in the PI number after the point when calculating the length of the meridian, we will be mistaken by several millimeters. And when calculating the length of the Earth’s orbit when rotating around the Sun (as is known, R = 150 * 106 km = 1.5 * 1014 mm), for the same accuracy it is enough to use the number PI with fourteen digits after the dot, and what’s there to waste - the diameter of our Galaxies are about 100,000 light years away (1 light year is approximately equal to 1013 km) or 1018 km or 1030 mm, and back in the 17th century, 34 digits of PI were obtained, which are excessive for such distances, and they are currently calculated to 12411 trillionth sign!!!
The absence of periodically repeating numbers, namely, based on their formula Circumference = Pi * D, the circle does not close, since there is no finite number. This fact can also be closely related to the spiral manifestation in our lives...
There is also a hypothesis that all (or some) universal constants (Planck’s constant, Euler’s number, universal gravitational constant, electron charge, etc.) change their values over time, as the curvature of space changes due to the redistribution of matter or for other reasons unknown to us.
At the risk of incurring the wrath of the enlightened community, we can assume that the PI number considered today, reflecting the properties of the Universe, may change over time. In any case, no one can forbid us to re-find the value of the number PI, confirming (or not confirming) the existing values.
10 interesting facts about PI number
1. The history of numbers goes back more than one thousand years, almost as long as the science of mathematics has existed. Of course, the exact value of the number was not immediately calculated. At first, the ratio of the circumference to the diameter was considered equal to 3. But over time, when architecture began to develop, a more accurate measurement was required. By the way, the number existed, but it received a letter designation only at the beginning of the 18th century (1706) and comes from the initial letters of two Greek words meaning “circle” and “perimeter”. The letter “π” was given to the number by the mathematician Jones, and it became firmly established in mathematics already in 1737.
2. In different eras and among different peoples, the number Pi had different meanings. For example, in Ancient Egypt it was equal to 3.1604, among the Hindus it acquired a value of 3.162, and the Chinese used a number equal to 3.1459. Over time, π was calculated more and more accurately, and when computing technology, that is, a computer, appeared, it began to number more than 4 billion characters.
3. There is a legend, or rather experts believe, that the number Pi was used in the construction of the Tower of Babel. However, it was not the wrath of God that caused its collapse, but incorrect calculations during construction. Like, the ancient masters were wrong. A similar version exists regarding the Temple of Solomon.
4. It is noteworthy that they tried to introduce the value of Pi even at the state level, that is, through law. In 1897, the state of Indiana prepared a bill. According to the document, Pi was 3.2. However, scientists intervened in time and thus prevented the mistake. In particular, Professor Perdue, who was present at the legislative meeting, spoke out against the bill.
5. Interestingly, several numbers in the infinite sequence Pi have their own name. So, six nines of Pi are named after the American physicist. Richard Feynman once gave a lecture and stunned the audience with a remark. He said he wanted to memorize the digits of Pi up to six nines, only to say "nine" six times at the end of the story, implying that its meaning was rational. When in fact it is irrational.
6. Mathematicians around the world do not stop conducting research related to the number Pi. It is literally shrouded in some mystery. Some theorists even believe that it contains universal truth. To exchange knowledge and new information about Pi, a Pi Club was organized. It’s not easy to join; you need to have an extraordinary memory. Thus, those wishing to become a member of the club are examined: a person must recite from memory as many signs of the number Pi as possible.
7. They even came up with various techniques for remembering the number Pi after the decimal point. For example, they come up with entire texts. In them, words have the same number of letters as the corresponding number after the decimal point. To make it even easier to remember such a long number, they compose poems according to the same principle. Members of the Pi Club often have fun in this way, and at the same time train their memory and intelligence. For example, Mike Keith had such a hobby, who eighteen years ago came up with a story in which each word was equal to almost four thousand (3834) of the first digits of Pi.
8. There are even people who have set records for memorizing Pi signs. So, in Japan, Akira Haraguchi memorized more than eighty-three thousand characters. But the domestic record is not so outstanding. A resident of Chelyabinsk managed to recite by heart only two and a half thousand numbers after the decimal point of Pi.
9. Pi Day has been celebrated for more than a quarter of a century, since 1988. One day, a physicist from the popular science museum in San Francisco, Larry Shaw, noticed that March 14, when written, coincides with the number Pi. In the date, the month and day form 3.14.
10. There is an interesting coincidence. On March 14, the great scientist Albert Einstein, who, as we know, created the theory of relativity, was born.